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Euclid to Einstein: Barker Builds Beautiful Geometry

Story posted October 12, 2005

Anyone who thinks geometry is merely the study of shapes, angles, and points on a plane has not encountered Mathematics Professor Bill Barker talking about the Erlanger Programm and transformational geometry:

"Geometric thinking is at the heart of many applications in science, in art, and at the heart of a huge amount of seemingly non-geometric mathematics," beams Barker. "Even space-time can be viewed as encoded by specific symmetry groups."

"Geometry is like a skyscraper," he says. "You keep building more and more and get this amazing intellectual universe. It's beautiful."

Anyone who thinks numbers and equations are the sole tools of geometry also is wrong: If you want to understand geometry, look to language. Words such as "symmetry," "equivalence," and "transformation" take on hugely complex meanings in a mathematical setting, but once described, they open up universes of thought.

Barker has had to have an artful grasp on both word and number over the past year. During his 2004-2005 sabbatical, he has been hard at work finishing up Volume I of Continuous Symmetry: From Euclid to Einstein, a new undergraduate textbook to be published in two volumes by the American Mathematical Society in 2006.

The result of nearly a decade of work with his co-author, Yale University's Roger Howe - a world-renowned mathematical researcher - the book promises to be one of the most comprehensive undergraduate treatments of the transformational approach to geometry.

Barker and Howe will present advancing levels of geometry, based on an advanced concept - but designed to be accessible to a wide audience. They cover geometric theory and application from its earliest Euclidean iteration as "geometry of the world" (geo is Greek for earth, and metron means measure), through more modern, abstract systems, including hyperbolic geometry, projective geometry and Lie theory. The book ends with a comparison of two different geometries of space-time that come from classical versus relativistic physics.

The unifying theme throughout this broad survey is the concept of transformational geometry - known as the Erlanger Programm - which Barker says "allows you to think of one geometry as 'sitting inside of' the other."

The Erlanger Programm was developed by the 19th-century German mathematician Felix Klein during a time when certain long-held conjectures about Euclidean geometry were shown to be false, opening up whole new concepts of geometry.

"This geometry may seem weird to a non-mathematician," concedes Barker, smiling. "But it makes perfect sense when viewed in the scheme of the Erlanger Programm."

"People started devising other geometries, such as hyperbolic, projective, and elliptic geometry," says Barker. "It was an exciting but confusing time in mathematics, and it spawned the question: What unites these various systems and makes them all geometry? What is geometry?"

Klein came up with one intellectual idea that encompassed all the various geometric systems, a formulation still without serious rival. Simply stated, he defined geometry as the study of properties of objects (tangible or abstract) that do not change under a selected group of movements, the "symmetries" of the geometry.

It's a deceptively simple statement.

"In classical geometry," explains Barker, "the symmetries - or 'rigid motions' - are simply descriptions of any way of moving a plane without changing the distances between any pair of points. You can flip, rotate, or translate the plane - moving it like a cardboard sheet across a desk - and you don't change the distances between two points.

"It's the combinations of these basic rigid motions that give us what we call the symmetries of Euclidean geometry."

Things start to get interesting, however, when the groups of symmetries change. The symmetries of a plane change considerably, for instance, if you allow magnifications by any fixed positive number. Such a magnification from a fixed point is known as dilation (or a contraction, if reduced by a positive number).

Those magnifications produce a group of "similarities" of the plane, a variant of Euclidean geometry known as similarity geometry - well known to high school students studying similar triangles.

A rendering of the Poincaré disk, which illustrates the standard model of hyperbolic geometry. Through the point P there are three (in fact, an infinite possible number of) hyperbolic lines that are parallel to line L (in red), meaning, they do not intersect L. This shows the failure of the Parallel Postulate in hyperbolic geometry. William Barker illustration.

One of the most beguiling geometries is what's known as hyperbolic geometry. "It's a form of geometry where parallel lines do something very strange," notes Barker. "In Euclidean geometry, if you are given a point and a line, there is exactly one line that passes through the point and is parallel to the original line. It is what's known as the parallel postulate.

"In the hyperbolic plane, there are an infinite number of such parallel lines. Those 'straight lines' curve and appear as half circles, but they satisfy all the axioms of Euclidean geometry except for the parallel postulate."

It was just this exception to the postulates of Euclidean geometry that unleashed the genie of modern geometry.

"This geometry may seem weird to a non-mathematician," concedes Barker, smiling. "But it makes perfect sense when viewed in the scheme of the Erlanger Programm. We simply have an unusual group of symmetries. In fact, the geometry of relativistic space-time is very much like hyperbolic geometry.

"Viewing geometry this way takes your thinking to another level of abstraction," says Barker. "You're not just thinking about triangles and circles as objects, you're now thinking of symmetries as objects, not just 'movements,' but abstract objects. You end up understanding geometry from different viewpoints. Geometric thinking is at the heart of many applications in science, in art, and at the heart of a huge amount of seemingly non-geometric mathematics. Even space-time can be viewed as encoded by specific symmetry groups."

Barker has been "test-driving" the book's chapters and lessons in his classes at Bowdoin, and reports that this teaching approach not only "works" with undergraduates, but that "students have come up with many improvements; they often have seen a better approach to a specific problem than I have.

"We hope the book will be widely adopted," adds Barker. "We have a serious goal: to make the Erlanger Programm the central organizing theme for undergraduate geometry wherever it's taught."

For Barker, it is something of a crusade: "Students often get turned off to mathematics because they don't see the big ideas. We are not always good at presenting them," he says. "Instead, we get sidetracked by computational details.

"In this book, the whole course is built around this huge intellectual idea: Geometry is the study of invariant properties of objects under a selected group of symmetries. What does that mean? How do all the various geometries fit this scheme? How do you 'do geometry' via transformations? The questions are fascinating, challenging, and unending!"